Structural controllability of general edge dynamics in complex network

Dynamic processes that occur on the edge of complex networks are relevant to a variety of real-world systems, where states are defined on individual edges, and nodes are active components with information processing capabilities. In traditional studies of edge controllability, all adjacent edge states are assumed to be coupled. In this paper, we release this all-to-all coupling restriction and propose a general edge dynamics model. We give a theoretical framework to study the structural controllability of the general edge dynamics and find that the set of driver nodes for edge controllability is unique and determined by the local information of nodes. Applying our framework to a large number of model and real networks, we find that there exist lower and upper bounds of edge controllability, which are determined by the coupling density, where the coupling density is the proportion of adjacent edge states that are coupled. Then we investigate the proportion of effective coupling in edge controllability and find that homogeneous and relatively sparse networks have a higher proportion, and that the proportion is mainly determined by degree distribution. Finally, we analyze the role of edges in edge controllability and find that it is largely encoded by the coupling density and degree distribution, and are influenced by in- and out-degree correlation.


Lemma 1
The minimum number of driver nodes need to fully control the nodal dynamics of L(G ′ ) is one if there is a perfect matching in its bipartite graph H(A). Otherwise, it equals the number of unmatched nodes with respective to any maximum matchings. In this case, the driver nodes are just the unmatched nodes.
Now we propose a framework to determine the minimum driver nodes and driven edges required to control GED. We divide H(G) into N matching blocks based on N switching matrices. The matching block corresponding to a switching matrix S v contains the incoming edge states and outgoing edges edge states of the node v, where the incoming edge states and outgoing edges edge correspond to the left and right nodes of the matching block, respectively. Meanwhile the independent free parameters in S v correspond one-to-one to the edges in the matching block.
Remark 1 Each matching block is independent, i.e., there is no edge linking the nodes from two different matching blocks. The left and right node sets of N matching blocks just cover all the incoming and outgoing edge states of G.
Then we give the lemma as follows.

Lemma 2
The matching block of S v contains unmatched nodes if and only if S v has no full row-rank, i.e., rank g (S v ) < k + v . The number of unmatched nodes in the matching block of S v is equal to k + v − rank g (S v ).
Proof 1 According to Remark 1, we can prove the lemma by one matching block and its corresponding switching matrix. We first prove the sufficiency. For a node v with k + v × k − v switching matrix S v , the left and right node sets of its matching block correspond to the number of columns and rows of S v , respectively. Any two matching edges in a matching block correspond to two independent free parameters with different rows and columns in S v . These two parameters are irreducible and will add 1 to the generic rank of S v , respectively. So the maximum number of matching edges of a matching block is equal to the generic rank of S v . When rank g (S v ) < k + v , the number of matching edges is less than the number of nodes in the right node set, resulting in unmatched nodes in the matching block. Meanwhile, the number of unmatched nodes in the matching block of S v is equal to Then we prove the necessity. When the matching block contains unmatched nodes, the number of matching edges is less than the number of nodes in the right node set. Since the maximum number of matching edges of a matching block is equal to the generic rank of S v , the matching block contains unmatched nodes that can deduce rank g (S v ) < k + v .
Combining Lemma 1 and 2, we can prove the main conclusions described by Eqs. (4) and (5). The proof process is as follows.
Proof 2 Eq. (2) indicates that the GED of a digraph G is equivalent to the nodal dynamics of its trimmed line graph L(G ′ ). They have the same state set and state matrix W . The equivalence shows that the driver nodes in L(G ′ ) correspond one-to-one with the driven edges in G. Applying the minimum input theorem to the nodal dynamics of L(G ′ ) gives us the bipartite graph H(G). Then the maximum matching method can determine the unmatched nodes in H(G), which correspond to the driver nodes required to control the nodal dynamics of L(G ′ ), and correspond to the driven edges required to control the GED of G. The maximum matching method determines the minimum number of unmatched nodes in H(G). Accordingly, the number of driven edges required to control the GED of G is also minimal. Meanwhile, Eq. (2) indicates that the driver node for controlling GED of G is the starting node of the driven edge. Therefore, we can determine the minimum number of driver nodes and driven edges required to control the GED of G.
We divide H(G) into N matching blocks based on N switching matrices. According to Remark 1, we can give the proof with one matching block and its corresponding switching matrix. According to Lemma 2, a matching block of S v contains unmatched nodes if and only if S v has no full row-rank. Therefore, a node v is the driver node if and only if rank g ( To ensure reachability, we randomly select a node in each full-rank component to apply the external input. Since the input matrix H of the GED is a diagonal matrix, the nodes to which the external input is applied are all driver nodes. In summary, the minimum number of drive nodes required to control the GED is According to Lemma 2, if a node v is the driver node, the number of unmatched nodes in the matching block of S v is equal to k + v − rank g (S v ). Therefore, the outgoing edge set of the driver node v contains k To ensure reachability, we randomly select an outgoing edge of the driver node in each full-rank component as the driven edge. In summary, the minimum number of driven edges required to control the GED is

Supplementary Note 2: Analytical results
The dependence of edge controllability on local network information allows us to derive analytical results in terms of the coupling density P and the joint degree distribution Firstly, we calculate, by statistical methods, the probability that a k + v × k − v switching matrix satisfies rank g (S v ) = k + v under the given P. Specifically, P ∈ [0, 1] quantifies the probability that an element in the switching matrix is an independent free parameter. For a given k + v × k − v all-zero switching matrix S v , we determine whether each element is 0 or an independent free parameter based on the given P. Then we calculate the generic rank rank g (S v ) of the switching matrix and judge whether it satisfies rank g (S v ) = k + v . Through 1000 such independent simulations, the probability in each independent simulation. The mean P M i, j of 1000 independent simulations is an estimate of the number of driven edges for the k + v × k − v switching matrix under the given P. Thirdly, we compute the proportion rank g (S v )/(k + v k − v P) of effective coupling in each independent simulation. The mean P E i, j of 1000 independent simulations is an estimated proportion of effective coupling for the k + v × k − v switching matrix under the given P. Finally, we calculate the proportion of three kinds of edges based on their judgment method in each independent simulation, and get the estimated proportions of critical P C i, j , ordinary P O i, j and intermittent P I i, j edges for the k + v × k − v switching matrix under the given P.
We neglect the possible presence of full-rank component, which is uncommon in directed networks and has little effect on N D and M D . Meanwhile, we assume that in-and out-degrees of each node are uncorrelated to offer analytical results. The divergent (k + v > k − v ) node must not satisfy rank g (S v ) = k + v , and the balanced (k + v = k − v ) and convergent (k + v < k − v ) nodes may satisfy rank g (S v ) = k + v . Therefore, the proportion n D of driver nodes is given as Then the proportion m D of driven edges is where the average degree is ⟨k⟩ = ⟨k − ⟩ = ⟨k + ⟩ = M/N. When P = 1, all the elements in the switching matrix of each node are independent free parameters, n D and m D reach the lower bounds. Their analysis results have been given in 3 . Specifically, the lower bound of driver nodes is n L D = 1 2 (1 − ∑ ∞ i=0 P ii ) and the lower bound of driven edges is m L D = 1 ⟨k⟩ ∑ ∞ i=0 ∑ ∞ j=1 jP i,(i+ j) . When P = 0, all the elements in the switching matrix of each node are 0, the n D and m D reach the upper bound. The upper bound of driver nodes is n U D = 1 − ∑ ∞ i=0 P i,0 . The upper bound of driven edge is m U D = 1. When P is given, the estimated proportion of effective coupling for the k + v × k − v switching matrix is P E i, j . Therefore, the proportion P E of effective coupling is When P = 1, the proportion of effective coupling is When P = 0, we have P E = 0.

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The outgoing edges of the nodes with k + v > k − v = 0 are all critical edges. Meanwhile, when P is given, the estimated proportion of critical edges for the k + v × k − v switching matrix is P C i, j . Therefore, the proportion m C of critical edges is Similarly, the proportion m O of ordinary edges is Then the proportion m I of intermittent edges is When P = 1, the proportion of critical edge is When P = 0, all the elements in the switching matrix of each node are 0, causing m C = 1, m O = 0 and m I = 0.
The ER networks are generated by static model 4 , where both the in-and out-degrees follow a Poisson distribution.
By substituting the Poisson distribution into Eq. (7), we obtain Similarly, the analysis result of m D is When P = 1, the lower bounds 3 are n L D = 1 2 (1 − e −2⟨k⟩ I 0 (2⟨k⟩)) and m L D = e −2⟨k⟩ ⟨k⟩ ∑ ∞ j=1 jI j (2⟨k⟩), respectively, where I a (x) is the modified Bessel function of the first kind. When P = 0, the upper bounds are n U D = 1 − e −⟨k⟩ and m U D = 1, respectively. The analysis result of P E is where HF[(a 1 , ...,  a p ), (b 1 , ..., b q ), z] is the generalized hypergeometric function. When P = 0, we have P E = 0. The analysis result of critical edge is Similarly, the analysis result of ordinary edge is Then the analysis result of intermittent edge is When P = 1, we have m C = e −<k> , m O = e −2⟨k⟩ ∑ ∞ j=1 I j (2⟨k⟩) and m I = e −2⟨k⟩ ⟨k⟩ ∑ ∞ j=1 (i + j)I j (2⟨k⟩) − e −⟨k⟩ . When P = 0, we have m C = 1, m O = 0 and m I = 0.
The EX networks with an exponential degree distribution are generated by configuration model 5 . Both the in-and out-degrees follow the same exponential distribution, which is where C = 1 − e −1/κ and κ = 1/log 1+⟨k⟩ ⟨k⟩ . By substituting the exponential distribution into Eq. (7), we obtain Similarly, the analysis result of m D is When P = 1, the lower bounds 3 are n L D = ⟨k⟩ 2⟨k⟩+1 and m L D = ⟨k⟩+1 2⟨k⟩+1 , respectively. When P = 0, the upper bounds are n U D = ⟨k⟩ ⟨k⟩+1 and m U D = 1, respectively. The analysis result of P E is . When P = 0, we have P E = 0. The analysis result of the critical edge is The analysis result of the ordinary edge is The analysis result of the intermittent edge is When P = 1, we have m C = 1 ⟨k⟩+1 , m O = ⟨k⟩(⟨k⟩+1) (2⟨k⟩+1) 2 and m I = ⟨k⟩ 2 (3⟨k⟩+2) (⟨k⟩+1)(2⟨k⟩+1) 2 . When P = 0, we have m C = 1, m O = 0 and m I = 0. The SF networks are generated by static model 4 . Both the in-and out-degrees follow a power-law distribution 6 , i.e., where Γ(x, y) is the incomplete Gamma function. Let δ denote [⟨k⟩(1−a)] . By substituting the power-law distribution into Eq. (7), we obtain

Supplementary Note 3: Real network analysis results
We substantiate how to derive theoretical predictions of real networks. Specifically, we insert the coupling density, probability P N i, j and degree distribution of a real network into Eq. (7) to predict the fraction of driver nodes n analytic D via n analytic D where P in (x) and P out (y) are the fraction of nodes in the real network with degree k − v = x and k + v = y, respectively. The theoretical predictions of the fraction of driven edges m